Question

How many times from 4 p.m to 10 p.m, the hands of a clock are at right angle?

• A. 11
• B. 6
• C. 9
• D. 10

Hint:

Remember, while calculating clock angles, consider the fact that right angles occur twice per hour in most cases, but always account for exceptions like when the hands are directly opposite to each other (i.e., at 180 degrees).

Answer 1

The Correct Option is A

A clock's hour and minute hands form a right angle when the angle between them is exactly 90 degrees. This occurs multiple times in each hour, specifically twice: once when the hour hand is ahead and once when it is behind the minute hand. However, due to the continuous movement of the clock's hands, the number of times they align at 90 degrees is not always straightforward.
From 4 p.m. to 10 p.m., we are looking at a span of 6 hours. For each hour, the hands of the clock will align at right angles at two different times. However, there is one crucial exception: at 6 p.m., the hands of the clock will be exactly opposite each other (i.e., at 180 degrees) and thus will not form right angles. Therefore, when calculating from 4 p.m. to 10 p.m., the hands of the clock form right angles 11 times in total.
To break it down further, the occurrences are:
- 4:00 - 4:32 (one occurrence)
- 5:00 - 5:27 (one occurrence)
- 6:00 - 6:32 (but this is an exception, no right angle)
- 7:00 - 7:27 (one occurrence)
- 8:00 - 8:32 (one occurrence)
- 9:00 - 9:27 (one occurrence)
Thus, the total number of occurrences from 4 p.m. to 10 p.m. is 11 times.